Aircraft: MEDIUM LARGE TRANSPORT; LOW WING; 2 TURBOJET ENG
Phase of flight: Initial Approach
Anomaly: Conflict Airborne Conflict; Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Published Material / Policy; Deviation / Discrepancy - Procedural Clearance

To this date I have not been given time to review the radar data or the voice tapes. The Operations Manager told me; the day after; that they would get me time to review and fill out ATSAP. The week after; a Front Line Manager removed me off position at the end of my shift (18 minutes at the end of a 10 hour day) and wanted to discuss the event. He offered me to look at the data and he wanted to talk about the incident. I told him we would need more than 18 minutes to review the data but I would be happy to discuss the incident.Aircraft X on Visual Approach to RWY XXL. Aircraft Y released off of ZZZ heading 220 degrees; climbing to 3000 feet. I vectored the Aircraft X tight on base to allow room for the military transport to turn downwind. When Aircraft X was about 10 mile base Aircraft Y was airborne and in a right turn to downwind.I noticed Aircraft Y turning eastbound toward Aircraft X. I tried to contact Aircraft Y twice with no response. I then quoted the military transport to Aircraft X to establish visual separation. Aircraft X did not see the military transport so I immediately turned Aircraft X heading 130 degrees in an attempt to keep him away from the military transport who at this time is eastbound; potentially Nordo; and climbing to 3000 feet. I called ZZZ tower to see if they still had Aircraft Y; then tried again to contact Aircraft Y. My concern was that the aircraft was Nordo and potentially going to turn back toward the southwest and I had to keep Aircraft X clear. I then called tower to see if they were talking to Aircraft Y and they responded 'yes.' I told them to turn Aircraft Y heading 270 degrees climbing to 3000 feet establish positive separation; still not knowing that the Aircraft Y had been cleared to land. As soon as I was able to issue that control instruction I got off the line and noticed that an aircraft was airborne on Runway XXR [at ZZZ1]. I immediately went back to ZZZ1 tower to stop the departure at 2000 feet below Aircraft X. Aircraft Z was through 2500 feet so I stopped him at 2500 feet. I then got off the line and vectored Aircraft X heading 180 degrees to ensure diverging flight paths between Aircraft Z and Aircraft X and climbed Aircraft X to 4000. Unfortunately circumstances did not allow for conversation or questions; nor did they allow for proper landline phraseology.Aircraft were coming together rapidly and action needed to be taken to keep them apart! My actions were predicated on a wayward aircraft and not knowing what that aircraft was doing. I only had time to react and not ask questions. Tower controller listen to the aircraft and respond appropriately. Had the ZZZ1 Tower controller listened to the Aircraft Y pilot who reported on frequency climbing to 3000; he may have questioned the pilot and either looked out the window or at the radar display to accurately realize what was going on! This could have been avoided! Apparently the military transport pilots have tried this before; going to ZZZ1 tower directly to minimize delay. Some of these pilots fly for [Aircraft X's carrier] also so they are well aware of procedures and traffic flows. This pilot was probably trying to beat the system. He picked an unfortunate time and caused a perfect storm! I cleared Aircraft Y to land on Runway XYR; thinking that he had been cleared for the Visual Approach by the approach controller. Aircraft X was inbound for the channel Runway XXL. I told Aircraft Y about Aircraft X; thinking that Aircraft X would either be maintaining visual separation or be restricted above Aircraft Y. Approach then called to ask if I was talking to Aircraft Y; and to issue a left turn and climb. I later realized that Aircraft Y had never checked in with approach; and the approach controller had no idea what Aircraft Y was doing. If an aircraft checks in that soon after departing ZZZ; then verify that they already spoke to approach control. TRACON Controller had an aircraft on approach to one airport and a second aircraft released off an adjacent airport flying to the same airport as the one on approach. The departure aircraft did not contact the TRACON Controller but immediately contacted the arrival Tower Controller. The TRACON Controller vectored the aircraft on approach away and attempted to get contact with the departure. The TRACON Controller discovered the departure was on Tower frequency and relayed control instructions to further separate the traffic. In the meantime the Tower departed another aircraft which was climbing in to confliction with the arriving aircraft being vectored for traffic. The Controller stopped this departure's climb; but not in enough time to prevent a loss of separation.